I used to have my main battling pet in my first pet slot, but since I also frequently use the zapper, I accidentally zapped him instead of my lab rat pet, since your first pet also shows up as the default for that. Is there a way to set a default battling pet so I can use him when I hit the "battle with defaults" button under each opponent without needing to keep him in my first pet slot?

Hungry pets or pets with 0 HP can't fight, perhaps you could try having your lab rat pet in the first slot but keep it hungry or with 0 HP and your battle pet in the second slot.